Proving that a certain region in $\mathbb{R}^2_{>0}$ has a bounded number of intervals when intersected with a line

Question

Let $R$ be a subset of $\mathbb{R}^2_{>0}$ such that the following conditions hold: $$ x < y < x(1 + \sqrt{1 + \frac{B}{A}}) $$ $$ y(Ay + Bx) \leq E $$ $$ C \leq y $$ $$ D \leq A y + B x $$ where $A, B, C, D, E$ are positive numbers. I would like to prove that there exists $N> 0$ (which does not depend on $A, B, C, D, E$) such that for any $x_0$ and $y_0$, $(\mathbb{R} \times \{ y_0\}) \cap R$ is a union of at most $N$ intervals and $(\{x_0 \} \times \mathbb{R}) \cap R$ is also a union of at most $N$ intervals. I think this is true.... I would appreciate a solution for this. Thank you.

Answer 1

$N=2$ suffices. We note that all of these constraints with the exception of $y(Ax+By)\leq E$ are linear constraints which determine a (closed or open) half-space. Let $S$ be the region of $\Bbb R^2_{>0}$ satisfying these constraints: it is convex, as it is the intersection of convex sets. Then the intersection of any line with $S$ is either empty or has exactly one connected component, by the definition of a convex set.

Next, the intersection of any line with $y(Ax+By)=E$ is either empty, one point, two points, or the entire line. In particular, this means the intersection of any line with the region $y(Ax+By)\leq E$ has at most two connected components, each of which is convex. It is now immediate to see that the intersection of any line with $R$ has at most two connected components.

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In general, one may use facts about semialgebraic geometry to get the existence such a bound without determining the explicit value. Consider a semialgebraic set $A\subset \Bbb R^m\times \Bbb R^n$, which we think of as a family of semialgebraic sets parameterized by $x\in\Bbb R^m$. By taking a finite cylindrical cell decomposition of $X$, one sees that there is a global bound on the number of connected components of $A_x:=A\cap (\{x\}\times\Bbb R^n)$ in terms of the global data of $X$. (For more on this, see any introductory text or survey of semialgebraic or o-minimal geometry - I like "A long and winding road to definable sets" by Denkowska and Denkowski and "An Introduction to Semialgebraic Geometry" by Coste, for instance.)

How this applies in your situation is that we can give a copy of $\Bbb R^8$ the coordinates $(A,B,C,D,E,a,b,c)$ and $\Bbb R^2$ the coordinates $(x,y)$ and consider the semialgebraic set given by the intersection of your defining equations involving $A,\cdots,E,x,y$ and the semialgebraic set given by $ax+by+c=0$, remove the fiber over $a=0,b=0$ while retaining semialgebraicness, and now we have a semialgebraic set $X\subset \Bbb R^8\times\Bbb R^2$ who's fiber over a fixed point in $\Bbb R^8$ is exactly the intersection you're interested in.